Mathematics of Materials

September 20-24, 2004

Speakers:

Kaushik Bhattacharya
Applied Mechanics & Mechanical Engineering
Division of Engineering & Applied Science
California Institute of Technology
bhatta@cco.caltech.edu
http://mechmat.caltech.edu
Biography

Masao Doi
Department of Applied Physics
Tokyo University
doi@cse.nagoya-u.ac.jp
http://www.stat.cse.nagoya-u.ac.jp/~masao/
Biography

Qiang Du
Department of Mathematics
Pennsylvania State University
qdu@math.psu.edu
http://www.math.psu.edu/qdu/
Biography

Chun Liu
Department of Mathematics
Pennsylvania State University
liu@math.psu.edu
http://www.math.psu.edu/liu/
Biography

Ellad B. Tadmor
Faculty of Mechanical Engineering
Technion - Israel Institute of Technology
tadmor@technion.ac.il
http://tx.technion.ac.il/~tadmor
Biography

The 2004-2005 IMA thematic program on "Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities" will begin with a tutorial on "Mathematics and Materials," during the week September 20-24, 2004. The tutorial week will consist of lectures by five distinguished researchers on background topics in methods to analytically and numerically address emerging modeling problems for materials. Applications will include multiscale methods for gels, liquid crystals, superconductivity, micromagnetics, elastomers, and crystalline solids.

The tutorial lectures are scheduled for 9-10 am, 10:30-11:30 am, 1:30-2:30 pm and 3:00-4:30 pm, on Monday through Friday of the week of September 20 to 24 of 2004.

Topics and Lecture Abstracts

Kaushik Bhattacharya (California Institute of Technology)

Biography: Kaushik Bhattacharya is a Professor of Mechanics and Materials Science at the California Institute of Technology. He received his Ph.D in Mechanics from the University of Minnesota in 1991 and his post-doctoral training at the Courant Institute for Mathematical Sciences during 1991-1993. He has held visiting positions at Cornell University, Heriot-Watt University (Scotland), Max-Planck-Institute (Leipzig, Germany), Cambridge University (England) and the Indian Institute of Science (Bangalore, India). He received the Young Investigator Award from the National Science Foundation (NSF) in 1994, Charles Lee Powell Award in 1997, the Young Investigator Prize from the Society of Engineering Science (SES) in 2003 and the Special Achievements Award in Applied Mechanics from the American Society of Mechanical Engineers in 2004. He is currently the Editor of the Journal of the Mechanics and Physics of Solids, and serves on the Editorial Board of three other journals. He has organized numerous international meetings including a four-month program at the Isaac Newton Institute, Cambridge, England on the mechanics of materials in 1999, and the recent SIAM Conference on the Mathematical Aspects of Materials Science in 2004. His research interest concerns modeling problems that arise in Materials Science, especially in the area of Active Materials.

Energy Minimization and Microstructure
Paper:   pdf

Abstract. There are numerous phenomena in materials science where fine-scale microstructure is the result of the material seeking to optimize multiple incongruent objectives. Examples include alloy phase segregation, martensitic phase transformation, nematic elastomers, ferroelectrics and faceting of crystalline surfaces. Further the ability of a material to form microstructure and to change its microstructure depending on the macroscopic boundary conditions endow the materials with unusual macroscopic behavior like the shape-memory effect, electrostriction and the liquid-like behavior of solids.

These series of lectures will describe selected examples of such phenomena and how such a phenomenon can naturally be modeled as a variational problem, specifically a minimization problem with non-convex energy density. It will show that microstructure arises as an inevitable consequence of such a variational problem and that nontrivial aspects of the microstructure can be predicted from such a formulation. Finally, it will introduce the notion of effective behavior, i.e., the overall behavior of the material after it has formed microstructure, how microstructure gives rise to very unusual effective behavior and how one can describe it without having to resolve every fine detail of the microstructure. These lectures are intended to be accessible to a broad audience with a balance between phenomena, modeling and mathematical analysis.

Masao Doi (Tokyo University)

Biography: Dr. Masao Doi is Professor of Computational Science and Engineering in Nagoya University. He was a Fellow of the Science Research Council at Cambridge University from 1976 through 1978. Professor Doi has received awards from the Polymer Society of the Japan and Rheology Society of Japan for his research on polymer dynamics and rheology. He is also the recipient of the Japan IBM Award of Science and doctor honoris causa of Katholic University Leuven, Belgium. His monograph, `The Theory of Polymer Dynamics` with Sam Edwards has become the standard reference on nonlinear rheology of flexible and rod-like polymers. Professor Doi received the Polymer Physics Prize in 2001 "For pioneering contributions to the theory of dynamics and rheology of entangled polymers and complex fluids."

Modeling of Gels (The Coupling Between Stress and Diffusion)

The lecture topics will be distributed as follows:

(1) What is a gel   Slides:   pdf
(2) Stress diffusion coupling: the phenomena and modeling  Slides:   pdf
(3) Stress-diffusion coupling in polymer solutions   Slides:   pdf
(4) Electro-responsive gels  Slides:   pdf

Abstract: A gel is an elastic object swollen by solvent, so the force acting on the gel is coupled with the diffusion of the solvent. The stress-diffusion coupling is seen commonly in everyday life (water coming out of a squeezed gel) and is also important in many chemical engineering processes, soaking, drying and sedimentation. The stress diffusion coupling is also important in the study of artificial muscles, where the deformation of the gel is controlled by an electric field. Professor Doi will present equations for the stress diffusion coupling for an ionic gel and discuss electro-chemical effects.

Qiang Du (Pennsylvania State University)

Biography: Qiang Du is a professor of mathematics at the Pennsylvania State University, University Park. He received his Ph.D. in 1988 under the direction of Max Gunzburger and then went on to do as a Dickson Instructor at the University of Chicago. He then served as assistant and associate professor in the Mathematics Department at Michigan State University from 1990-1996, before holding professorships at Iowa State University and the Hong Kong University of Science and Technology. Qiang Du's research interests span many areas but include numerical algorithms, partial differential equations, parallel and scientific computation and applications to the physical sciences. He is, in particular, internationally recognized as one of the world's leading researchers in the area of Ginzburg-Landau theory and superconductivity.

Mathematical Models of Superconductivity, an Introduction

Abstract: Superconductivity is one of the grand challenges identified as being crucial to future economic prosperity and scientific leadership. In recent years, the analysis and simulations of various mathematical models in superconductivity have attracted the interests of many mathematicians all over the world. Their works have helped us to understand the intriguing and complex phenomena in superconductivity.

With the recent award of the Nobel Prize in Physics, a renewed attention has been focused on theoretical foundations of superconductivity, for example, the popular Ginzburg-Landau theory was proclaimed as "being of great importance in physics ...". There are new and unresolved mathematical challenges be explored further. In this tutorial, we will briefly review the physical background of some interesting problems related to superconductivity, in particular, the problem of quantized vortices. Various mathematical models ranging from microscopic BCS theory to the macroscopic critical state models will then be described with the meso-scale Ginzburg-Landau model being our emphasis. Some recent analytical and numerical results will be surveyed. Connections to other relevant problems such as the vortices in Bose-Einstein condensation will also be discussed.

Chun Liu (Department of Mathematics Pennsylvania State University)

Biography: Dr. Chun Liu is an Associate Professor of Mathematics in the Pennsylvania State University, University Park. He received his Ph.D. in Mathematics in 1995, from the Courant Institute of Mathematical Sciences, New York University. He was a postdoctoral research fellow in the Department of Mathematics, Carnegie Mellon University, Pittsburgh, during the academic year 1995-1996. In the following year, he held the Richard Duffin Visiting Assistant Professor position in the same department. Chun Liu research interests center around partial differential equations and calculus of variations, with applications to complex fluids, liquid crystals and polymeric materials, mixtures and interfaces, magneto-hydrodynamics and electro-kinetic flow, elasticity and grain growth. He is a very active researcher and speaker.

Variational Approaches in Complex Fluids

Abstract: Complex fluids such as polymeric solutions, liquid crystal solutions, pulmonary surfactant solutions, electro-kinetic fluids, magneto-rheological fluids and blood suspensions exhibit many intricate rheological and hydrodynamic features that are very important to biological and industrial processes.

The most common origin and manifestation of anomalous phenomena in complex fluids are different "elastic" effects. They can be the elasticity of deformable cells, elasticity of the molecule alignment in liquid crystals, polarized colloids or multi-component phases, elasticity due to microstructures, or bulk elasticity endowed by polymer molecules in viscoelastic complex fluids. The physical properties are purely determined by the interplay of entropic and structural intermolecular elastic forces and interfacial interactions. These elastic effects can be represented in terms of certain internal variables, for example, the orientational order parameter in liquid crystals (related to their microstructures), the distribution density function in the dumb-bell model for polymeric materials, the magnetic field in magneto-hydrodynamic fluids, the volume fraction in mixture of different materials etc. The different rheological and hydrodynamic properties can be attributed to the special coupling between the transport of the internal variable and the induced elastic stress. From the point of the view of the energetic variational formulation, this represents a competition between the kinetic energy and the elastic energy.

In these lectures, I will study three different but related types of problems to illustrate this unified energetic variational approach. All the systems are related and have common structures. However, each one posses its own distinct features (difficulties). I will present some modeling and analytical results, as well as those problems that remain to be solved.

Ellad B. Tadmor (Faculty of Mechanical Engineering, Technion - Israel Institute of Technology)

Biography: Dr. Ellad B. Tadmor is a senior lecturer in the Department of Mechanical Engineering at the Technion - Israel Institute of Technology in Haifa, Israel. Dr. Tadmor's research focuses on understanding material response from fundamental principles rather than phenomenology. He studies microscopic processes that lead to macroscopic phenomena such as fracture and plasticity using atomic-scale modeling and multiple-scale techniques. Prior to his current position, Dr. Tadmor was a postdoctoral research fellow in the Division of Engineering and Applied Sciences at Harvard University working with Prof. Efthimios Kaxiras on incorporating ab initio models into multi-scale methods. In 1996 Dr. Tadmor received his Ph.D. in Engineering from Brown University in Providence, RI. His doctoral research with Prof. Michael Ortiz and Prof. Rob Phillips focused on the development of the Quasicontinuum Method, a mixed continuum and atomistic formulation for describing the mechanical response of materials at the atomic scale. Dr. Tadmor has received a number of awards including several Technion Awards for Excellence in Teaching, the Salomon Simon Mani Award for Excellence in Teaching, and the Materials Research Society (MRS) Graduate Student Award for his Ph.D. work.

Multiple-Scale Modeling of Materials using the Quasicontinuum Method

Tentative titles for the three lectures are:
1. Materials and Multiple Scales
2. The Theoretical Foundations of the Quasicontinuum Method
3. Quasicontinuum Applications

Abstract: Atomistic and continuum methods alike are often confounded when faced with mesoscopic problems in which multiple scales operate simultaneously. In many cases, both the finite dimensions of the system as well as the microscopic atomic-scale interactions contribute equally to the overall response. This makes modeling difficult since continuum tools appropriate to the larger scales are unaware of atomic detail and atomistic models are too computationally intensive to treat the system as a whole.

We present an alternative methodology referred to as the "quasicontiuum method" which draws upon the strengths of both approaches. The key idea is that of selective representation of atomic degrees of freedom. Instead of treating all atoms making up the system, a small relevant subset of atoms is selected to represent, by appropriate weighting, the energetics of the system as a whole. Based on their kinematic environment, the energies of individual "representative atoms" are computed either in nonlocal fashion in correspondence with straightforward atomistic methodology or within a local approximation as befitting a continuum model. The representation is of varying density with more atoms sampled in highly deformed regions (such as near defect cores) and correspondingly fewer in the less deformed regions further away and is adaptively updated as the deformation evolves.

The method has been successfully applied to a number of atomic-scale mechanics problems including nanoindentation into thin aluminum films, microcracking of nickel bicrystals, interactions of dislocations with grain boundaries in nickel, junction formation of dislocations in aluminum, cross-slip and jog-drag of screw dislocations in copper, stress-induced phase transformations in silicon due to nanoindentation, polarization switching in ferroelectric lead-titanate and deformation twinning at aluminum crack tips. An overview of the methodology and selected examples from these applications will be presented.

LIST OF CONFIRMED PARTICIPANTS

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